Simplify square root of 18a^2*(4 square root of 3a^2)

$\sqrt{18 a^{2}} \cdot (4 \sqrt{3 a^{2}})$

Step 1

Rewrite using the commutative property of multiplication.

$4 \sqrt{18 a^{2}} \sqrt{3 a^{2}}$

Step 2

Rewrite $18 a^{2}$ as ${(3 a)}^{2} \cdot 2$ .

Factor $9$ out of $18$ .

$4 \sqrt{9 (2) a^{2}} \sqrt{3 a^{2}}$

Rewrite $9$ as $3^{2}$ .

$4 \sqrt{3^{2} \cdot 2 a^{2}} \sqrt{3 a^{2}}$

Move $2$ .

$4 \sqrt{3^{2} a^{2} \cdot 2} \sqrt{3 a^{2}}$

Rewrite $3^{2} a^{2}$ as ${(3 a)}^{2}$ .

$4 \sqrt{{(3 a)}^{2} \cdot 2} \sqrt{3 a^{2}}$

$4 \sqrt{{(3 a)}^{2} \cdot 2} \sqrt{3 a^{2}}$

Step 3

Pull terms out from under the radical.

$4 (3 a \sqrt{2}) \sqrt{3 a^{2}}$

Step 4

Simplify the expression.

Multiply $3$ by $4$ .

$12 (a \sqrt{2}) \sqrt{3 a^{2}}$

Reorder $3$ and $a^{2}$ .

$12 a \sqrt{2} \sqrt{a^{2} \cdot 3}$

$12 a \sqrt{2} \sqrt{a^{2} \cdot 3}$

Step 5

Pull terms out from under the radical.

$12 a \sqrt{2} (a \sqrt{3})$

Step 6

Multiply $a$ by $a$ by adding the exponents.

Move $a$ .

$12 (a \cdot a) \sqrt{2} \sqrt{3}$

Multiply $a$ by $a$ .

$12 a^{2} \sqrt{2} \sqrt{3}$

$12 a^{2} \sqrt{2} \sqrt{3}$

Step 7

Multiply $12 a^{2} \sqrt{2} \sqrt{3}$ .

Combine using the product rule for radicals.

$12 a^{2} \sqrt{3 \cdot 2}$

Multiply $3$ by $2$ .

$12 a^{2} \sqrt{6}$

$12 a^{2} \sqrt{6}$

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